Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
3.2.1
![**Problem Statement:**
Find \(\vec{r}''(t)\), given \(\vec{r}(t) = \langle te^{2t}, -\cos(5t + 5), 3t \ln(-t) \rangle\).
**Solution Steps:**
1. **Determine the first derivative \(\vec{r}'(t)\):**
To find \(\vec{r}'(t)\), take the derivative of each component of \(\vec{r}(t)\).
2. **Calculate the second derivative \(\vec{r}''(t)\):**
Differentiate each component of \(\vec{r}'(t)\) to find \(\vec{r}''(t)\).
**Errors in Provided Solution:**
The proposed solution for \(\vec{r}''(t)\) was:
\[
\langle -6e^{-2t}, 4\cos(2t), -3t^2 \rangle \quad \text{(incorrect)}
\]
The incorrect components should be recalculated by taking the second derivatives of the original function's components:
- The first component of \(\vec{r}(t)\) is \(te^{2t}\).
- The second component of \(\vec{r}(t)\) is \(-\cos(5t + 5)\).
- The third component of \(\vec{r}(t)\) is \(3t \ln(-t)\).
**Note:**
Each of these calculations needs to be corrected to find the accurate \(\vec{r}''(t)\).
This task requires careful reevaluation of the differentiation process to correct these errors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b463452-b960-4b00-bce0-3a69d9f467e2%2F4041a1bb-8a4c-447c-afa8-a2e7aad10fa6%2F9epwky_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \(\vec{r}''(t)\), given \(\vec{r}(t) = \langle te^{2t}, -\cos(5t + 5), 3t \ln(-t) \rangle\).
**Solution Steps:**
1. **Determine the first derivative \(\vec{r}'(t)\):**
To find \(\vec{r}'(t)\), take the derivative of each component of \(\vec{r}(t)\).
2. **Calculate the second derivative \(\vec{r}''(t)\):**
Differentiate each component of \(\vec{r}'(t)\) to find \(\vec{r}''(t)\).
**Errors in Provided Solution:**
The proposed solution for \(\vec{r}''(t)\) was:
\[
\langle -6e^{-2t}, 4\cos(2t), -3t^2 \rangle \quad \text{(incorrect)}
\]
The incorrect components should be recalculated by taking the second derivatives of the original function's components:
- The first component of \(\vec{r}(t)\) is \(te^{2t}\).
- The second component of \(\vec{r}(t)\) is \(-\cos(5t + 5)\).
- The third component of \(\vec{r}(t)\) is \(3t \ln(-t)\).
**Note:**
Each of these calculations needs to be corrected to find the accurate \(\vec{r}''(t)\).
This task requires careful reevaluation of the differentiation process to correct these errors.
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning